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Embedded Fluid Structure Interaction Technique for Thin Def

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Masoud Davari

on 26 September 2013

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Transcript of Embedded Fluid Structure Interaction Technique for Thin Def

Embedded Fluid Structure Interaction Technique for Thin Deformable Structure
STATE OF THE ART
Masoud Davari

Advisor: Dr. Pooyan Dadvand
Co-Advisor: Dr. Riccardo Rossi


INTRODUCTION AND MOTIVATION
FSI
Interaction of movable, rigid or deformable structures with internal or surrounding fluid flow
Influence of fluid and structure dynamics to each other:

Structure deform under the action of the fluid stresses
This deformation, in turn, changes the boundary conditions of the fluid flow (Fluid follows the structure displacement )
significant challenge:
large movement, or the large deformation of structure due to serve fluid stresses
geometrical complexity
Re-mesh processing is needed
expensive
difficulty in mapping the state variables from the old mesh to the new mesh
The approaches to the field of Fluid Structure Interaction for handling moving interfaces of an elastic solid structure can be subdivided into two classes,
conforming mesh
meshes are conformed to the interface where the physical boundary conditions are imposed. In this class due to the large deformation and/or movement of the solid structure, re-meshing is needed as the solution is advanced
non-conforming mesh
meshes are not conformed to the interface. As a result, the fluid and solid equations can be conveniently solved independently from each other with their respective grids, and re-meshing is not necessary.
conforming mesh methods
Arbitrary Lagrangian Eulerian method (ALE)
An ALE method allows arbitrary motion of grid/mesh points with respect to their frame of reference by taking the convection of these points into account
This method goes back to early works like (Belytschko et al. 1980; Belytschko and Kennedy 1978; CEL 1964; Codina et al. 2009; Hirt et al. 1974; J. Donea et al. 1977; J. Hughes et al. 1981)
conforming mesh methods
Particle Finite Element Method (PFEM)
The key feature of the PFEM is that the equations of motion of the fluid domain are formulated in a Lagrangian framework. In other word PFEM is a particular class of Lagrangian formulation aiming to solve problems involving the interaction between fluids and solids a unified manner.

The PFEM method for the solution of FSI problems is recently used by (Aubry et al. 2004; A Larese, R Rossi 2008; Oñate 2004; Oñate et al. 2003 ; Oñate and Periaux 2006 ).
advantage:
The convective term in the momentum conservation equation disappears.
disadvantage:
Re-meshing at each time step is required.
non-conforming mesh methods
Immersed Boundary Method
- A finite difference grid for the fluid domain
- An immersed set of non-conforming boundary points that are mutually interconnected
by an elastic law
- Solid boundary interacts with the fluid by means of local body forces applied to
the fluid at the position of the structure
- IB has been successfully applied in many application fields (Dillon and Fauci
2000; Gilmanov and Sotiropoulos 2005; Jung 1999; Zhu and Peskin 2002)
Fictitious Domain Method
non-conforming mesh methods
- Evolved from the field of finite elements
- Coupling is established by constraining fluid and rigid body at the interface using a
(distributed) Lagrange multiplier
- Lagrange multipliers consist of adding new equations to the global system of
equations that enforce the boundary conditions
- New unknowns (the Lagrange multipliers) need also to be added to the problem
advantage:
mesh updating not required
disadvantage:
the system of equations becomes larger, and the space for the Lagrange multipliers has to be carefully chosen so that the final formulation is stable
non-conforming mesh methods
Extended Immersed Boundary Method
and
Immersed Finite Element Method
- describe the solid using the finite element method
- describe the fluid is formulated using a finite difference or finite element method
- The coupling is performed using the discrete dirac delta functions that find
their origin in the mesh-less Reproducing Kernel Particle Method (RKPM).
- Within these methods, the conservative and computationally stable coupling of solid and
fluid can be realized by higher-order schemes, increasing the numerical effort.
In these method that mentioned in non-conforming mesh class ,the interaction of the fluid and the solid is taken into account through a force term which appears either in the strong or in the weak form of the flow equations.
Here, in this work, our aim is to propose a method so that the structure and fluid component
is coupled by interface conditions not by imposing the force term
some key approaches related to our proposal :
eXtended Finite Element Method (XFEM):
Coupling Strategies (partitioned approach)
The extended finite element method (XFEM) is frequently used in order to incorporate discontinuous solution properties into the approximation space. Discontinuities inside elements, as they e.g. occur in two-phase/free-surface flows with implicit interface descriptions, can thereby be accounted for appropriately. In the extended finite element method (X-FEM), a standard space based finite element approximation is enriched by additional (special) functions using the framework of partition of unity. In other word XFEM is a numerical method, based on the Finite Element Method (FEM), that is especially designed for treating discontinuities.
In the X-FEM, the finite element mesh need not conform to the internal boundaries
Former researcher
multi-fluid (Chessa and Belytschko 2003; Coppola-Owen and Codina 2005; Fries and Belytschko 2010; Henning and Thomas-Peter 2011; Sven and Arnold 2007)
FSI
(Motasoares et al. 2006; Sawada and Tezuka 2011)
Coupling of multi-physics problems for instance FSI can be simulated in two approaches
monolithic approach
- Both the flow equations and structural equations are solved simultaneously
partitioned approach
- Requires a code developed for this particular combination of physical problems
- Flow equations and structural equations are solved separately
- Preserves software modularity
- Are typically explicit and staggered
- Advantage: allows solving the flow equations and the structural equations with
different , possibly more efficient techniques which have been developed specifically
for either flow equations or structural equations
What is our proposed approach
Here we proposed an Embedded FSI technique for Thin Deformable Structures using FEM with enrichment method based on XFEM
what is new in our work is that likes all non-conforming mesh methods, capable handle large deformations/motion of the structure because fluid and solid meshes are generated independently from each other with this difference that dislike immersed boundary and fictitious methods, the interaction of the fluid and the solid is taken into account through common interface conditions not by applying a force term
Important points of this method are as follows
• non-conforming mesh method can reduce human and computational costs and difficulties of the mesh
generation
• This method has a potential capability of handling the geometrical complexity of FSI problems, most of which
is caused by large deformation and translation or motion of structures.
• Partitioned approach for Coupling fluid-structure, allows solving the flow equations and the structural
equations with different, possibly more efficient techniques which have been developed specifically for either
flow equations or structural equations.

OBJECTIVES
The objective is to analyze Fluid deformable Structure Interaction with large structural deformations/motion without re-meshing process

To achieve the objective, we propose a non-conforming mesh method based on eXtended Finite Element Method (X-FEM) that is especially designed for treating discontinuities in velocity and pressure in fluid arises from structure. This method can reduce human and computational costs and difficulties of the mesh generation. At the end we will compare the results obtained by our method and the structure-conforming mesh methods.

METHODOLOGY
Fluid
Structure
XFEM
FSI
Governing equation of fluid
conservation of momentum

Mass conservation
If we assume Newtonian behavior, through the constitutive equation it is possible to relate the fluid stress tensor to the fluid velocity

Therefore the Navier-Stokes equations is

Boundary conditions
Governing equation of structure
Where:
boundary conditions
- Partitioned approach is used
Fluid Structure Interaction
- The influence of the structure on the fluid
- The influence of the fluid on the structure
eXtended Finite Element Method
- The Extended Finite Element Method (XFEM) is a numerical method, based on the Finite Element Method (FEM), that is especially designed for treating discontinuities
- Discontinuities are generally divided in strong and weak discontinuities.
- Strong discontinuities are discontinuities in the solution variable of a problem
- Weak discontinuities are discontinuities in the derivatives of the solution variable
In this study, two methods related to XFEM are used
An enrichment method using additional degree of freedom
A modified XFEM method in which no additional degrees of freedom are incorporated.
consider a piecewise linear approximation field
Enrichment method using additional degree of freedom
defined in a cut triangular element
the element can then be split into two sub-elements
and
weak discontinuous
strong discontinuous
Modified XFEM (without additional degree of freedom)
In this case no additional degrees of freedom are incorporated but the shape functions are modified so as to capture discontinuities
NA(A) = 1 NB(A) = 0 NC(A) = 0
NA(B) = 0 NB(B) = 1 NC(B) = 0
NA(C) = 0 NB(C) = 0 NC(C) = 1
NA(P+) = 1 NB(P+) = 0 NC(P+) = 0
NA(P-) = 0 NB(P-) = 1 NC(P-) = 0
NA(Q+) = 1 NB(Q+) = 0 NC(Q+) = 0
NA(Q-) = 0 NB(Q-) = 0 NC(Q-) = 1

Values at the vertices of the sub-triangles
PRELIMINARY RESULTS
Heat transfer problem
Stokes problem
We consider a heat transfer problem so that one thin fissure is embedded into a heat flow domain
The model equation for transient heat transfer by conduction
boundary conditions:
weak form of transient heat transfer eguation
where:
In finite element, the temperature is approached by the discretized form:
Matrix form of weak equation:
where:
The system has to be integrated in time, therefore we have
Enrichment method
For the elements which are intersected by the interface, approximation function consists of a standard finite element (FE) part and the enrichments parts
To define weak equation we decompose the discrete problem by using test functions from the linear, weak discontinuous and jump part, respectively
Matrix form:
where
Now, using the fact that the enrichment functions are local to each element, we eliminate
and
the elementary level before final assembly as follows
Modified XFEM method:
In this case for those elements that are cut by the interface, the classical conforming space, locally modified to accommodating discontinuities at the (given) interface
Note:
This method does not introduce any additional degrees of freedom
Results:
We consider the heat flow domain (0,6)×(0,6) with one thin fissure
Finite element space with conforming mesh
Enrichment method
Modified XFEM
Contour lines of Temperature
Deformable Structures:
An important field of FSI problems involves single or multiple thin and deformable structures like membranes
Extremely light.
The geometry depends strongly on the given stress distribution
Increasing lightness and slenderness brings along a higher susceptibility to flow-induced deformations and vibrations when exposed to the fluid load
In civil engineering this exemplifies in wind-induced effects on thin shells and membrane structures.
These wind effects can define the decisive design loads and therefore require an in-depth analysis. This phenomenon is called aeroelasticity in cases of an interaction between structure and wind. It can occur on constructions such as towers, high-rise buildings, bridges, cable and membrane roofs, etc
What we will to introduce:
Here, our project aim is to analyze Fluid deformable Structure Interaction with large structural deformations/motion without re-meshing process
The Stokes problem in differential form:
Weak formulation of Stokes problem
Where
Discrete problem for Galerkin Method
Matrix form of discrete problem
Consistently stabilized methods for the Stokes equations: (GLS)
Galerkin terms
Stabilization terms
Stabilization terms
residual of momentum equation
test function terms
It is then easy to see that the discrete problem is equivalent to a family of linear algebraic systems of the form
Where
Enrichment method
,

,

,

,

,

,

we decompose the discrete problem by using test functions from the standard and enrichment parts
Matrix for of discrete problem
Re-write the matrix of discrete problem
Static condensation
Modified XFEM method:
In this case for those elements that are cut by the interface, the classical conforming space, locally modified to accommodating discontinuities at the (given) interface
Note:
This method does not introduce any additional degrees of freedom
Results:
We consider the fluid domain (0,6)×(0,6) with one thin structure
TIMETABLE

showing contour lines of Pressure
showing contour lines of Velocity in X direction
showing contour lines of Velocity in Y direction
Courtesy of Daniel Baumgartner and Johannes Wolf
Conforming mesh
Enrichment Method
Modified XFEM
Conforming mesh
Enrichment Method
Modified XFEM
Conforming mesh
Enrichment Method
Modified XFEM
The material time derivative becomes a simple partial derivative with respect to time, such that
Structural acceleration
Cauchy stress tensor
Structural density
The structure is described using a Lagrangian description
Structural body forces
Full transcript